The Rational Expectations Hypothesis: A New Formalization

Copyright @ IFAC Computation in Economics. Finance and Engineering: Economic Systems. Cambridge. UK. 1998

The Rational Expectations Hypothesis: A New ' Formalization

with mean zero and variance dl. l Assuming that the model explaining the variable Yt+ 1 can be put in the form 2

Marco P. Tucci, Universita di Siena, Facolta di Giurisprudenza, Piazza S. Francesco, 7 - 53100 Siena, Italy Incorporating raiional expectations in a dynamic linear econometric model requires either to estimate the parameters of agents' objective functions and of the random processes that they faced historically (Hansen and Sargent, 1980) or to use a Fair and Taylor (1983) type procedure to determine the expected values of the endogenous variables. Sometimes the Blanchard and Kahn (1980) method is used to reduce -a first-order system with rational expectations to a form without expectations. In this paper it is shown that a model consistent with this hypothesis (Chow, 1983) is equivalent to the original model with time-varying parameters. Abslrucl:

Copyright

with cO' PO' PI' ... , KO' Kl' ... , constant coefficients and where the sums are infinite,3 its conditional expectation given the informalion al time L-l is v = -t+lIt-l

cO+P2 ut _I + .. ·+1C0 \+ IIt-l +K 1X t1t _I +K 2 X t _I + .. ·· (3) Subtracting (3) from (2), and taking into account that \ is defined in such a way that Xtlt - 1=xt' 4 gi ves

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~ 19981FAC

Keywords:

Rational expeclations, time-varying parameters, Kalman Filter.

1 INTRODUCfION Incorporating the rational expectations hypothesis (REi"l) in a dynamic linear econometric model requireseilher to estimate "the parameters of agents' objective functions l!nlj nf the random processes that they faced historically" (Hansen and Sargent, 1980) or to use a Fair and Taylor (1983) type procedure to determine the expected values of the endogenous variables, i.e. the states, in the future periods. Sometimes the Blanchard and Kahn (1980) method is used to reduce a first-order system with rational expectations to a form without expectations. In this paper a new econometric formalization of the REH is presented. Starting out with Chow (1983), in Section 2 it is shown ~hat the usual scalar model consistent with the REH is equivalent to the original model with time-varying parameters. Then a system of simultaneous equations is considered (Section 3). The identification problem, both for the single equation and the simultaneous equations cases, is discussed in Section 4. Section 5 is devoted to the estimation problem. The main conclusions are summarized in the final section. 2 THE USUAL RATIONAL EXPECf ATIONS MODEL: A SINGLE EQUATION Chow (1983, p. 355) considers a simple scalar model of the form (1)

where Yt+ IIt-l is the expected value of Yt+ 1 given all the information available at time t-l, Y is the actual t value of the endogenous variable at time t, x is the t

exogenous variable known at the end of period t-l and Ut is a serially uncorrelated random disturbance

347

-1 d with Po="( an et+ 1=\+ I-Xt+llt- l the error in forecasting xt+1 one period ahead . Under these

assumptions model (1) is implied by

Equation (5) "is a dynamic model explaining Y + 1 t which is consistent with the original model .. . (i.e. Eqt. (1» but does not utilize any expectation variables. It has two more parameters ... (PI and KO) ... than the original model. If only model ... (1) ... is given, one is free to choose the values of these parameters and thus produce many stochastic models in the form of ... (5) ... which are consistent with the specification ... (1)" (Chow (1983. p. 355-356».5 Therefore "just to say that Yt depends on what people expect its value will be in 't+ l' and on other factors is not a complete theory until one specifies how the expectation Yt+ I',t- 1 is formed . ... Simply

I The assumption of a normally distributed error term is not necessary in this discussion. However it is very useful in the estimation part to test hypothesis, to derive confidence intervals for the unknown parameters and to obtain the ~1a'(imum Likelihood estimators. 2 See Chow (1983). Taylor (1977) and Muth (1961) use the same general form but consider a case without exogenous variables. 3 As noticed in Chow (1983, p. 356), Eqt. (2) is justified because any linear model explaining .Y by Y - , ... . x ' t t I t xt _I ' ... and Ut can be put in this. form after repeated substitutions for the lagged ys when the model is stable. 4 See Chow (1983, p. 355). 5 Taylor (1977) has only one extra parameter because his model does not include the exogenous variable xl'

postulating rational expectations is not sufficient to complete the theory" (Chow (1983. p. 361».

uncorrelated with the forecast error terms associated with other exogenous variables and serially ' . 2 2 2 Thr d WIth uncorreIate vanance crk et=X k tcrk e' ere,ore 22 2/2 I ' . 2 t IS eac h 'I1 k t has vanance crk T{Y llC k 0 crk et interesting to notice that all the parameters associated with the truly exogenous variables have exactly the same form. they follow a MA(m-l) process. even though they do not necessarily ha\.'e the same coefficients.

An alternative way to write (5) is

x. .

(6)

with P1t~(POUt+ ((l+P 1)u t and P2t=1-lCo (e t+ /\). This means that the time-varying intercept follows an MA( I) model and the slope is a random parameter. I.e.

i3 1t-f3 1 = -

Pou t+ 1- (1+p I )u t =v t +

~2t-1 = - ICo(et + I /xt ) = 'I1 t

e

eV t _1

3 THE USUAL RATIONAL EXPECTATIONS MODEL: A SYSTEM OF SIMULTANEOUS EQUATIONS Consider nm\' a system of simultaneous equations such as

(7a) (7b) -1

with v t = (- po) ut + 1' = (l+p l )/ Po' Po = Y Furthermore if the forecast error on the exogenous variable et+ I is assumed unbiased and serially

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with Yt the Gxl vector of endogenous variables.

. h vanance ' 2 2 cr26 uncorre Iated WIt cret=x. e • 'I1 t possesses . 2/X.2 . . andh as vanance cr 2 =IC 2 cret the same propertIes 11 O Concluding the model with rational expectations (1) can be rewritten as a time-varying parameter model. namely a model with a time-varying intercept. PI l'

Yt+ Ilt-1 the vector of expectations formed at time t-

and a time-varying parameter associated with the exogenous variable. P2 t'

coefficients of dimension GxG, GxG and GxK

In general when the ex.pected value of a variable m periods ahead is needed and K-I truly exogenous variables appear in the relevant equation. Eqt. (1) becomes

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1. x t the Kx 1 vector of exogenous variables wi th "I"

as first element. Ut the vector of serially uncorrelated random disturbances with mean vector 0 and covariance matrix l:u and ri' rand B the matrices of

respectively. Pre-multiplying (11) by r 1 yields

with 9 =r l r , II=r -I B and vt=rlu Assuming, r l 1 as in the scalar case, that the endogenous vector can be modeled as

(8)

(13)

which can be rewritten in a time-varying parameters form as

with R j , Kj' v t and x t arrays of dimension GxG, GxK, Gxl and Kxl respectively, the difference between the actual value Yt+ I and its expected value at time t-l is Pit-PI =v t+el v t- I+ .. . +e mv t-m m-I ~t-j3k=xk t'l1 k t+ xk ti~l~k i'l1 k t-i

(lOa) (lOb)

with R =I.7 Therefore Eql (12) is implied by o

for k=2 ..... K where ~ktICk/lCk 0 for j=l . .... m-I

Yt+9IYt+I+IIXt

and Tlk t=-Y lICk o(e k t+ I/x k t)' The forecast error

=(1+91R1)v t +91v t+ 1+91KOe t + I

term associated with the k-th exogenous variable. ~ t+ I' is assumed unbiased. contemporaneously

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which can be rewritten in a time-varying parameter formal as

6 As explained in a previous footnote. the assumption of a normally distributed forecast error term is not necessary at this point even though it may be very useful in the estimation part.

7 See Chow (1983. p. 356).

348

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The system of simultaneous equations (15) can be rewritten as

with the time varying intercept and slopes in the g-th equation, at time t, defined as

{ G G }jG

= 1t - v +L v. Le ... Le .v. I gl g Ij=JJ Ij=1 glPIJ j=1 glll+1 for g=l, ... , G (17a)

*

}

G

(l7b)

4 THE IDENTIFICATION PROBLEM In Ihis section the identification problem is considered: namely are the parameters, included the extra parameters needed to obtain a unique solution to the REH model, identified? The case of a single equation involving the expectation of tomorrow's value of the endogenous variable corresponds to equation (1) which can be rewritten as equation (5) with two extra parameters to ensure uniqueness or, alternatively, as equation (6) with the time-varying parameters. In Eqt. (6) the intercept is represented by a simple MA( 1) model with one equation, e which can be consistently estimated from the correlogram, and one unknown PI because

A final remark concerns the identification of the extra parameters included in the Rand K matrices in a s\'stem of simultaneous equations. Chow (1983, p. 3'56-357) does not say anything about the shape (i.e. the nonzero elements) of the Rand K matrices. When the\' haye to be estimated however a problem of non-u~iqueness, or observational equival~nce, exists. Therefore they need to be correctly restncted in order to ensure •uniqueness '.11

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and PO=y-l . Therefore the extra parameter is identifiable. As far as the parameter associated with the exogenous variable in (6) is concerned the situation is less promising. The error term TIt has

/i

.8 This means that there is variance (J2=!C 2 (J2 11 0 e t -. one estimable quantity, namely the portion of the error term variance associated with the exogenous variable

c?e

with n, the matrix of reduced form parameters associated with the exogenous variables appearing in equation (12), containing the mean of these reduced form parameters and Et' the stochastic part of the parameters, including the extra parameters needed for a unique solution to the rational expectations models. The identification problem in this case is identical to that for a system of simultaneous equations with a disturbance"' term generated by a MA process and heteroskedastic. 10 Using a result shown in Kelejian (1974) the necessary and sufficient condition for the identifiability of the structural parameters in Eql. (19) is identical 'to that relative to the fixed coefficient version of the model. Therefore the complete identification problem, including both the structural parameters of system (12) and the extra parameters appearing in the moving average processes, can be divided into two parts. The identification problem of the structural parameters of Eql. (12) is the same as in the fixed parameter case and the identification of the extra parameters is possible, provided a sufficient number of observations is available, in the way discussed abo,-e.

1t gkl =1tk-ektILe ' !C' k g +j=IgII for g=l, ... , G, k= 2, ... , K

YI+9I y t+I +(n+E t )x t ==y t +91y 1+ l+nxt+Et=O (19)

5 ESTINlATING A SINGLE EQUATION: A 'THREE-STEP' PROCEDURE The direct estimation of equation (6) with the coefficient of Jt+ v 1 set equal to y 1 or, after

(J~, and two unknown parameters, !Co and

normalizing the coefficient of yt to minus one,

It is therefore impossible to distinguish one from

the other. A possible way out is to set KO= 1 and use . . 29 the abo"e equatton to estimate (Je Ut' as

the parameter associated with the exogenous

variable follows an MA(m-l). In this case however there are m unknowns. i.e. KO' Kl .... , Km_I' and m-I

8 For the problem of heteroskedasticity see Judge et a\. (1988. p. 436). 9 When an m-periods ahead expectation is involved the time-~' ar\'ing intercept is described by an MA(m) model. In this' case there are m equations involving m consistently estimable quantities from the correlogram of consistently estimated residuaIs, namely SI' S2' ....

equations 4> l' .... cl> m-I' As before one way to proceed is to set KO=I, and treat it as a numeraire, and use the estimate of the variance of "\ to 'derive information about the variance of the forecast error associated with the exogenous \. ariable. 10 Harvey and Phillips (l979) discuss the single equation case. 11 See the Appendix for an example of this procedure.

Srn' and m unknown parameters PI' P2 , ... . Pm' The estimate of the variance of v t can be used to derive information about the variance of the original error term 349

*

Y{'~Y lYt+l+~I+~2\+Ut'

*

(20)

(21)

for t;::K+I •...• T

vt=-Y IPOut+I'

and theirvariances f t=z;P tlt-l zr If the error terms are

9=(1+)'I Pl)/y 1PO and "t=-YllCo(et+l/xt)' using ordinary least squares (OLS) leads to inconsistent 1 estimates because plimT- (y t+ 1 );t(}. 12 This is due

independently and identically distributed normal . these quantities can be used to evaluate the prediction error decomposition of the likelihood function which looks like 16

to the fact that the error term is serially correlated because of multiperiod expectations 13 and it is correlated with Yt+1 defined as in (4).

10gL(y)= -( 1I2)(T-K)(log21t+logcr }-( 112) k log f t v t=K+l

with

Ut=Vt+6vt_l+Xt"t

where

U;

2

-2 T

parameters

2

-( 1I2)a L Ut If vt=K+I

In this section a 'three-step' estimation procedure is suggested first. Then a full ML estimator is derived embedding the KF recursions in a numerical optimization procedure. The former can be viewed as a straightforward generalization of the 'two-step' estimation procedure suggested in Harvey and Phillips (1979, p. 54). The first step consists in the consistent estimation of the parameters in (20) . Therefore the use of an instrumental variable (IV) approach is suggested. 14 Then the associated, consistent, residuals are used to find a consistent estimate of the covariance matrix and, in the final step, the model (20) is cast in state space form and the KF is run . This procedure will yield a feasible generalized least square (GLS) estimator for the mean of the parameters and a minimum mean square linear estimator (MMSLE) for their stochastic part conditional on the estimates of the hyperstructural

T

(22)

2 2 conditional on 8 and the ratio cricrv. The ML 2 estimator of cr ' conditional on 8 and the ratio v

ili. 11 v

ma\' be written in terms of standardized

.

prediction errors

(23)

2 2 \\'here v=
v

Therefore given the initial 'three-step' estimates for

e, cry2 and aT)2 (Harvey (1981, p. 104». 1::>-

"

.,

.

cr';'la~)'. say ",+. a set of T-3 residuals is 11 v obtained and the likelihood function (22) can be computed. Selecting a new set of values for ~e ., 2 * a~/crv)" say "', according to a numerical

",:::(8

As a by product of the computation of the GLS estimator the KF produces T-K, in this case T-3, prediction errors

optimization procedure a new value for the likelihood function is derived and the procedure is continued until convergence is attained. 18 If the ini tial estimators are consistent then under certain regularity conditions the final estimators are asymptotically distributed as the ML estimators. 19

12 See among the others Cuthbertson et al. (1992, p. 163).

13 See Cuthbertson et al . (1992, p. 170) for the case in which the structural errors Ut are serially correlated. 1~ Another way of approaching the problem of estimating (22) is to lag it one period (Chow (1983, p. 366» and to estimate -I * Yt=-(1 I) [yt-l-j31~Xt-l-~-I] which is a model with an autocorrelated error term and a lagged variable. As in the text the use of an IV approach is needed. 15 When the error term is normally distributed the KF recursions produce a GLS estimator for the mean of the parameters, which is asymptotically efficient when the hyperstructural parameters are consistently estimated, and a ~fMSE for their stochastic part conditional on the estimates of the hyperstructural pammeters (Harvey and Phillips (1979, p. 53». The estimator proposed in the text is similar in spirit to the 2S-2SLS estimator proposed in Cumby et al. ( 1983) and used in Cuthbertson (1990) and, when the error term is normally distributed. it is asymptotically more efficient than the estimator generated from the generalized method of moments (GMM) approach as noticed in Cuthbertson et al. (1992, p. 168-9).

16 This is the conditional likelihood when 8 is treated as random (Haryey (1991, pp. 130-133 and pp. 381-386». In the case in which 8 is unknown but fixed the summations run from 1 to T. The conditional likelihood is discussed also in Harvey (1981. p. 206). 17 See Haryey (1981. p. 198). 18 For a short survey of a number of different approaches to numerical optimisation see Judge et a!. (1985. p. 951). 19 See e.g. Harvey (1981 , p. 17) and Judge et al . (1985. pp. 196-205). When the normality assumption is dropped the estimator of the parameter vector is consistent and asymptotically normally distributed Judge et al . (1988, pp. 354-355). 350

6. ESTIMATING A SYSTEM OF SIMULTANEOUS EQUATIONS The 'three-step' procedure proposed in the previous section can be easily generalized to handle systems of simultaneous equations. As shown in Section 3, after some manipulation this model can be rewritten in a time-varying parameter format, i.e.

estimator of the covariance matrix and to find consistent estimates of the unknown parameters. The estimates associated with the constant term can be used to find consistent estimates of the parameters 9. , characterizing the time-varying intercept, and of I

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varying parameters associated with the truly exogenous variables in equation g, and of the ... ariances of the forecast error terms relative to these variables. As before using the estimates of the hyperstructural parameters obtained in step one and two the KF prediction and updating formulae can be appJied.21 The KF recursions yield the GLS estimator of S and the MMSLE of ~ conditional on

The g-th equation of this system, normalized so that the coefficient of y.,ot is minus one, can be rewritten as

=X 1t +Y 9 _ X Y Yg g gt 1+1 gig=< g 1+1

g) ( 91tgt}

Z ~

g

the variance of the error term of equation g. The estimates associated with the exogenous variables can be used to find consistent estimates of the parameters CPk·Ig for k=2, .. " Kg . characterizing the time-

..,

g gt(25)

1g

the estimates of hyperstructural parameters. with y 0 ' Y t+ I .,

0

eo

and Xg arrays of dimension Txl , When the disturbances Ut and the forecast errors et+ 1 are assumed normally distributed the limited information maximum likelihood (LIML) estimators of the parameters can be obtained. Given the initial

TxG t+ I g and TxK g respectivelv• containing the observations on the endogenous variable under consideration, future endogenous variables in the different periods and exogenous variables, respectively, included in the g-th equation and 1t t'

'three-step' estimates for v=<9

g

say '11+, the KF produces as a by product a set of T K residuals and the likelihood function can be computed. As in Section 5, selecting a new value for

9 1g the column vectors of unknown coefficients corresponding to the g-th row of

nt

and 9

1 respectively.20 Given that the time-varying vector of coefficients of the exogenous variables can be expressed as

'11, say 'If* , according. to a numerical optimization procedure a new value for the likelihood function is deri \·ed and the procedure is continued until convergence is attained. Again, if the initial estimators are consistent then under certain regularity conditions the final estimators are asymptotically distributed as the ML estimators.

(26)

Eqt. (25) takes the form y

= z S + DX ; g g g g g

with Z

g

and

g

7 CONCLUSION In this paper it has been shown that an econometric model estimated with the REH (Chow , 1983; Cuthbertson et al., 1992) is equivalent to the original model with time-varying parameters. This makes possible to keep the present format for most of the currently estimated models and it does not affect their identification. The parameters, and hyperparameters, of the model can be estimated with a 'three-step' procedure or maximizing the likelihood function. The former uses the KF recursions and produces consistent and asymptotically efficient estimators. The ML estimator is derived embedding the KF in a numerical optimization procedure. The new formalization, moreover, does not require to actually forecast the future values of the exogenous variables and it allows the possibility to test, rather than imposing. the validity of the rational expectations hypothesis. All these advantages seem to encourage its use in empirical works.

(27)

~=(1t'

g

9 10 ')' eo

of dimension

Tx(Kg+G +g 1 ) and (Kg+G +g 1 )xl, respectively, • t t DXg =diag(x g' x X of dimension TXTKg

i

with

2g, ... , Tg)

x:Jg the jth row of the X g matrix and

~ =(;'1 '

g

cr; ~a: ... ~ ~a:)',

g

r' , ... , r' )' the TKg vector of di sturbances. ~g '7fg As in the single equation case the direct application of OLS to (27) leads to inconsistent estimators. Therefore regress Y t+ 1 g on a set of variables Ag including all the predetermined variables in the complete system, say X, and those selected from y _ I), Y-2' ... , X_I and obtain the IV estimator of Sg' The residuals associated with this consistent estimator can be used to construct a consistent

21 The only remaining problem is to find suitable starting values. i.e. for the KF recursions (Harvey (1991. p. 121».

20 See Chow (1983, p. 370).

351

REFERENCES Blanchard, O.J. and C.M. Kahn (1980). The solution of linear difference models under rational expectations Econometrica, 48, 1305-1311. Chow, G.C. (1983). Econometrics. McGraw Hill, New York, NY. Cuthbertson, K., S.G. Hall and M.P. Taylor (1992). Applied econometric techniques. Philip Allan, Hemel Hempstead. Fair, R.c. and J.B. Taylor (1983). Solution and maximum likelihood estimation of dynamic rational expectations models. EC01wmetrica, 51, 1169-1185. Hansen, L.P. and T.J. Sargent (1980). Formulating and estimating dynamic linear rational expectations models, J. of Econ. Dyn. and COllt., 2, 7-46. Harvey, A.C. (1981). Time Series Models. Philip Allan, London. Harvey, A.C. (1991). Forecasting Structural Time Series Models and the Kalmall Filler. Cambridge University Press, Cambridge. Harvey, A.C. and G.D.A. Phillips (1979) . The estimation of regression models with autoregressive moving average disturbances. Biom., 66, 49-58. Kelejian, H. (1974). Random parameters in a simultaneous equation framework: identification and estimation. Ec01wmetrica, 42, 517-529. Judge, G. G., W. E. Griffiths, R. C. Hill, H. Lutkepohl and T. L. Lee (198S). The Theory and Practice of Ecollometrics (2nd Ed.). Wiley, New York. NY. Judge, G. G., R. C. Hill, W. E. Griffiths, H. Lutkepohl and T. L. Lee (1988). llllroductioll to the Theory and Practice of Econometrics (2nd Ed.). Wiley, New York, NY. Muth; J.F. (1%1). Rational expectations and the theory of price movements. Ec01wmetrica, 29, 315335. Taylor, J.B. (1977). Conditions for unique solutions in stochastic macroeconomic models with rational expectations. Econometrica, 45, 1377-1386.

APPENDIX Let q=l, Yt' Y1+1It-l and x t be defined as in (11) with 0=2 and K=3 and

(A 1)

This implies that

with FI and F2 linear in the arguments. Consequently the conditional means are of the form Y1 t+lIt-I=E( YI t+I Ut-I)=G I (Y2t+2It_l' X2t+lIt-l) (A3)

Y2 1+lIt-l=E( Y2 t+1 Ut_I)=G2(YI 1+2It-l' ~ 1+1It-l)' Rewriting (A3) it yields

YI 1+lIt- I=H1(Yl t+3It-I' x3 t+2It-l' x2 t+llt-l) (A4) Y2 1+1It-l=H2(Y2 t+3It-l' x2 t+2It-I' x3 t+llt-l) and when the endogenous variables can be modeled as \" -jC

PlY'I t- 1+'" +PPI·y·I t -PI.+K I X 1 t+'" +KI I·X1I· t+Vt (AS) for i=l. 2. 3. with p.I the maximum lag for endogenous variable i and lj the number of exogenous variables appearing in the i-th equation. Eql (A4) can be rewritten as

\"

=

(A6a)

- I t+llt-I L 1(y 1 t - l' ...• YI t -PI' X3 1+ 21 t - l' x2 1+ 11 1- 1)

Y2 t+ Ilt-1 = (A6b) L 2 (Y2 t-I' ...• Y2 t-P2' X2 1+2It-]' ~ t+llt-l) because E(v jtUt-l )=0. Assuming that the dynamic model is stable. i.e. (A7) Yj tlt-I= rl\'j t_l+ r2 Vj t-2 + •.. + kOxt + k 1:'
352

Y1 H IIt-I

=

Ql(v 1 t-I' vI t-2"'" Y::! HlIt-I

(ASa) X3

t' ~ t-1' ... , X 2 t' X2 t-I" " )

=

Q2(v 2 t-I' V2 t-2"' "

(A8b) X2

t' X2 t-I"'" ~ t' ~ t-I" " )'

Substituting (A8) into (A3) and this into (A2) ~ields

ylt=ZI(V I t-l' ... , x 2 t' ... , ~ t' ... ) y:h =Z2(v 2 t-I' ... , x2 l' ... , ~ t' ... )

(A9a)

(A9b)

Therefore the additional matrices RI and Ko needed to have unique sol utions to the REH model should be designed as22

(A 10)

22 The zero elements in the first column of KO are due to the presence of the intercept.

353